$\begingroup$If I'm not mistaken the only cross term contributing to the cross-variation process are the $dW_t$ terms, then (assuming your differentials are correct, and no correlation) $d[U,Z](t)=sin(2W_t)e^{W_t}dt$$\endgroup$
– zebullonJun 14 '15 at 3:21

$\begingroup$Thanks @zebullon - that makes sense to me. I forgot to put another part of the question in my post - is the quadratic variation of $Z_t$ equal to $(sin(2W_t))^2 dt$ ?$\endgroup$
– ShazJun 14 '15 at 3:50

$\begingroup$@Shaz Which definition of (cross)variation do you use?$\endgroup$
– sazJun 14 '15 at 5:50

1

$\begingroup$Recall that if $dU=AdW+Bdt$ and $dV=CdW+Ddt$ then by definition $d[U,V]=ACdt$. For example, $d[Z,Z]=\sin^2(2W)dt$. (But watch out for your $dU$, you computed $d(e^W)$, not $d(e^{W^3})$.)$\endgroup$
– DidJun 14 '15 at 20:40